Question

[BùiNgọcTháiHà16052010]

a)     Tìm số nguyên x,y biết: ( x - 3 ) . ( y + 1 ) = 15

b) Cho M = 1+ 3+3² + 3³ + 3⁴ + …+ 3⁹⁹ + 3¹⁰⁰ . Tìm số dư khi chia M cho 13, chia M cho 40 .

Answer 1

[Yen Nhi]

Answer:

a) 

Có:

\(15=1.15=\left(-1\right).\left(-15\right)=3.5=\left(-3\right).\left(-5\right)\)

x-3	1	15	-1	-15	3	5	-3	-5
y+1	15	1	-15	-1	5	3	-5	-3
x	4	18	2	-12	6	8	0	-2
y	14	0	-16	-2	4	2	-6	-4

b) 

Có:

\(M=1+3+3^2+3^3+3^4+...+3^{98}+3^{99}+3^{100}\)

\(=\left(1+3\right)+\left(3^2+3^3+3^4\right)+...+\left(3^{98}+3^{99}+3^{100}\right)\)

\(=4+3^2.\left(1+3+3^2\right)+...+3^{98}.\left(1+3+3^2\right)\)

\(=4+3^2.13+3^{98}.13\)

\(=4+13.\left(3^2+...+3^{98}\right)\)

Vậy M chia 13 dư 4

Có:

\(M=1+3+3^2+3^3+3^4+...+3^{99}+3^{100}\)

\(=1+\left(3+3^2+3^3+3^4\right)+\left(3^5+3^6+3^7+3^8\right)+...+\left(3^{97}+3^{98}+3^{99}+3^{100}\right)\)

\(=1+3.40+3^5.40+...+3^{97}.40\)

\(=1+40.\left(3+3^5+...+3^{97}\right)⋮40\)

Mà \(40\left(3+3^5+...+3^{97}\right)⋮40\)

Vậy M chia 40 dư 1